Problem: $\dfrac{ -8g + h }{ -2 } = \dfrac{ -10g - 9i }{ 8 }$ Solve for $g$.
Solution: Multiply both sides by the left denominator. $\dfrac{ -8g + h }{ -{2} } = \dfrac{ -10g - 9i }{ 8 }$ $-{2} \cdot \dfrac{ -8g + h }{ -{2} } = -{2} \cdot \dfrac{ -10g - 9i }{ 8 }$ $-8g + h = -{2} \cdot \dfrac { -10g - 9i }{ 8 }$ Multiply both sides by the right denominator. $-8g + h = -2 \cdot \dfrac{ -10g - 9i }{ {8} }$ ${8} \cdot \left( -8g + h \right) = {8} \cdot -2 \cdot \dfrac{ -10g - 9i }{ {8} }$ ${8} \cdot \left( -8g + h \right) = -2 \cdot \left( -10g - 9i \right)$ Distribute both sides ${8} \cdot \left( -8g + h \right) = -{2} \cdot \left( -10g - 9i \right)$ $-{64}g + {8}h = {20}g + {18}i$ Combine $g$ terms on the left. $-{64g} + 8h = {20g} + 18i$ $-{84g} + 8h = 18i$ Move the $h$ term to the right. $-84g + {8h} = 18i$ $-84g = 18i - {8h}$ Isolate $g$ by dividing both sides by its coefficient. $-{84}g = 18i - 8h$ $g = \dfrac{ 18i - 8h }{ -{84} }$ All of these terms are divisible by $2$ Divide by the common factor and swap signs so the denominator isn't negative. $g = \dfrac{ -{9}i + {4}h }{ {42} }$